A unified input-output approach for networked control problems with decentralized and selfish optimality

In this paper, we take an input-output approach to enhance the study of cooperative multiagent optimization problems that admit decentralized and selfish solutions, hence eliminating the need for an interagent communication network. The framework under investigation is a set of n independent agents coupled only through an overall cost that penalizes the divergence of each agent from the average collective behavior. In the case of identical agents, or more generally agents with identical essential input-output dynamics, we show that optimal decentralized and selfish solutions are possible in a variety of standard input-output cost criteria. These include the cases of ℓ₁, ℓ₂, ℓ∞ induced, and H₂ norms for any finite n. Moreover, if the cost includes non-deviation from average variables, the above results hold true as well for ℓ₁, ℓ₂, ℓ∞ induced norms and any n, while they hold true for the normalized, per-agent square H₂ norm, cost as n→∞. We also consider the case of nonidentical agent dynamics and prove that similar results hold asymptotically as n→∞ in the case of ℓ₂ induced norms (i.e., H∞) under a growth assumption on the H∞ norm of the essential dynamics of the collective.